The nanometer conductive film diffraction properties

Vulf Borisovich Orlenson, Sergey Aleksandrovich Zuev, and Vladimir Victorovich Starostenko*

V.I. Vernadsky Crimean Federal University, 295007, Simferopol, Russia

Abstract. In this article the numerical simulations and experimental

studies of microwave electromagnetic wave interaction with micro- and

nanoparticles of different sizes with their arbitrary location on the substrate

are carried. The optical coefficients (reflection, transmission and

absorption) and their dependence on the metallized nanocoating fill factor

and thickness were obtained.

1 Introduction

Conductive films are an inalienable element of electronic devices and applications made

using film technology. They are widely used in various fields, such as products and radio

devices shielding from external electromagnetic field exposure, infrared sensing, and solar

energy storage and in other branches of science and technology.

The properties of conductive films substantially depend on various parameters, in

particular, on their thickness, which determines the mechanisms of electromagnetic field

energy conversion into other types of energy. The main interaction mechanism of incident

radiation with a conductive coating is ohmic losses, which convert the electromagnetic

energy into thermal energy. The extension of thermoelectric processes in conductive films

is strongly influenced by their spatial and ohmic heterogeneity, in particular this relates to

contact pads, the burning of which is the main cause of integrated circuits (ICs) failures. In

[1,2] the film thickness dependences of the reflection coefficients for different conductive

materials were deeply studied. Starting from a certain film thickness (~5 – 7 nm), for most

conductive materials, a rapid increase of the reflection coefficient R was observed. Such

reflection behavior occurs due to the conductivity change and the formation of a reflective

layer. The waveguide experimental studies in [3] give a more complete picture of the

optical coefficients, where for an aluminum film thickness of the absorption

coefficient is . Film thickness rising leads to a conductivity increase as a result the

reflection grows while the absorption decreases. Also, the optical coefficients are frequency

independent.

This paper presents the theoretical study of the conductive thin films absorbing

properties behavior depending on metallized layer surface relief in the microwave range.

Theoretical studies are carried out by solving the EM wave scattering problem through the

numerical method of rigorous coupled wave analysis (RCWA) and analytical expressions

of the Fresnel-Airy model [4].

*Corresponding author: starostenkovv@cfuv.ru

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© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the CreativeCommons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/).

2 Conductive thin film analysis

In articles [5,6,7,8], experimental and theoretical studies of thin metal films absorbing

properties were carried out in the long-wavelength part of IR spectrum and the microwave

range. Irradiated by microwave EM waves conductive film of about 3 to 7 nm thickness

range, exhibits quite significant absorbing properties. A further film thickness growth leads

to increasing reflection. In particular interest are the [5.8] papers, in which the analytical

expressions for conductive films optical coefficients were obtained in the considered range.

These expressions have an interesting property, they are independent of the incident

radiation frequency, and besides the absorption maximum of 50% is achieved under

condition (1).

where – conductive film sheet resistance, – film conductivity – film thickness and

– free space impedance ( ). The theoretical derivations described above were

carried out under the assumption of conducting plane-parallel plate model with smooth

boundaries. Maxwell’s equations were solved in three layers and then stitched together at

the boundaries between them. The first and the third layer describe the free space regions

while the second layer between them describes the thin film region – the conductive layer.

Such a problem statement can also be considered using the Fresnel – Airy model [4], the

analytical expressions for which describe the complex reflection (2) and transmission (3)

coefficients. В этой модели показатели преломления взяты такие же, как и в работах

[5,8]. In this model, the refractive indices are taken the same as in [5,8], so the optical

coefficients are still frequency independent.

.

where , – complex reflection coefficients between the layers 1 and 2, 2 and 3, with

, – reflection and transmission coefficients, respectively,

– absorption coefficients. At normal incidence , where –

is wave vector at free space, – conductive film thickness, – refractive index.

The dielectric relative permittivity of conductive region is derived by , where

, – incident radiation angle frequency , – absolute dielectric constant in

vacuum and – magnetic relative permittivity ( ). In practice, the absorption

maximum of a thin conductive (aluminum) film obtained by magnetron sputtering is

observed at thicknesses from 5 to 10 nm [3,9]. However, if the conductivity of crystalline

aluminum equals to (or other crystalline metals with of the order of

) would be taken as the parameter , then the maximum absorption coefficient

will be at thickness [10,11], which does not coincide with the

experimental data. Using crystalline conductivity is not reasonably, since as a result of

sputtering techniques an amorphous conductive material is deposited on the substrate

surface. According to condition (1), the film conductivity for the considered thickness

should be of the order of , which may well correspond to the conductivity of

amorphous aluminum.

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Fig. 1. Optical coefficients dependence on the film thickness obtained by 1D model simulation.

3 Diffraction properties theoretical studies of conductive nanofilms

The properties of conductive films are associated not only with their thickness, but also with

the physical processes, which are inextricable linked with the characteristics of the substrates

on which they are deposited, forming metal-dielectric structures (MDS). The one-

dimensional formulation of EM wave scattering problem made it possible to obtain the

condition of reaching the absorption maximum (1). However, the real MDSs surface

topography studies by atomic and tunneling microscopy showed that such a problem

statement does not allow spatial and ohmic inhomogeneities analysis of the conductive layer

in terms of their contribution to the metal film absorbing properties. To solve this problem, the

MDS optical characteristics should be carried out using a 3D electromagnetic film model.

To calculate the optical coefficients, the numerical method of rigorous coupled wave

analysis (RCWA) was applied [12]. This method solves Maxwell’s equations in Fourier-

space. To study multilayer optical systems with complex relief, in the transverse direction

of each layer the eigenvalue problem is solved, after which the results are stitched together

at the boundaries of all layers. The 2D dielectric constant function of each layer should be

represented by a high-resolution real-space discrete grid, and then transformed in reciprocal

space (Fourier-space) (4).

where – transverse wave vector components, – spatial harmonics along

and axes respectively, – grid discrete points.

In this work, the S-matrices [13] method is used to carry out the procedure of solution

stitching and optical coefficients calculating. To calculate the optical coefficients, all the

considered energy contributions of the reflected and transmitted spatial harmonics should

be summed (5), (6).

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where – reflection coefficient, – transmission coefficient и – absorption coefficient.

The 3D model of the conductive film deposited on the substrate surface is presented in

layered 2D periodic lattice form with and periods along the corresponding axes and

. The unit cell of the conducting film is described by the 2D function , the real part

of which is shown in fig.2a, 3a. The incident EM wave is linearly polarized, and its electric

field vector is directed along the larger diagonal of the elliptical conducting particle. For the

same reasons as in the case of a 1D model, the conductivity of a metal particle is

. The substrate layer thickness is , with a relative dielectric

constant . Based on fig.1 at a thickness , in which the maximum

absorption should be achieved the influence of conductive layer geometry on the

optical characteristics can be studied. In the RCWA it is very important to look for

convergence, since the conservation of power will always be obeyed even considering just

one spatial harmonic, therefore, to obtain correct results, it is necessary to take into account

such number of spatial harmonics at which the changes in optical coefficients will be

insignificant. To solve the current problem the real-space grid resolution

was chosen. Fig. 2 shows optical coefficients plots for an array of periodically located elliptical

conducting particles. Such an MDS does not have continuous conduction channels;

therefore, taking into account a greater number of spatial harmonics the transmission

coefficient , fig.2b. In this case, the convergence of the results is achieved with a

large number of spatial harmonics.

Fig. 2. Metallized surface consisting of periodically arranged elliptical conductive particles:

a – unit cell dielectric constant plot, b – optical coefficients plot.

An increase elliptical particle size till it crosses the unit cell boundary leads to the

formation of continuous conducting channels, fig.3a. In this case, the transmission

coefficient decreases to about 65% while the absorption coefficient increases to 30%,

fig.3b. For such an MDS, the results converge with a much smaller number of spatial

harmonics, .

Based on the obtained calculations, it can be concluded that the absorbing properties of

a metal nanofilm strongly depend on the spatial distribution of its conductive particles,

although their linear dimensions and the distance between them are much smaller than the

incident wavelength, fig.4. The absorption maximum is achieved only with the formation of

continuous conductive channels.

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Fig. 3. Metallized surface consisting of periodically arranged intersecting elliptical conductive

particles: a – unit cell dielectric constant plot, b – optical coefficients plot.

Fig. 4. Spatial harmonics power composition studies. Transmission and reflection coefficient

dependence on spatial harmonics. The main energy of microwave radiation is contained only

in the main spatial harmonic.

4 Conclusion

The two description approaches of optical coefficients calculation are presented for metal-

dielectric structures with aluminum nanofilms. Numerical calculations show a strong

dependence of microwave range optical coefficients behavior on the conductivity, the

thickness and the spatial distribution of the conductive material on the substrate surface.

This type of EM wave scattering is realized due to structure transition from dielectric (the

film is absent) to a conducting film that short-circuited the space.

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